Joe Shipman said:
...For the category of sets I know how to prove
Schroder-Bernstein but I don't know a "categorial" proof...
Which inspired me to spend the better part of a day trying to find one,
and failing. I am leaning now toward thinking that there is no really
nice "categorical" proof of Schroder-Bernstein in the category of sets
without using the axiom of choice. My main reason for thinking this is
that the category of sets without choice is pretty close to the category
of partially ordered sets, and Schroder-Bernstein fails in that category
(the intervals (0,1) and [0,1] are mutually embeddable but not order
isomorphic). One reason for taking this analogy seriously is that if we
isolate all of the nice categorical properties characterizing the
category of sets, we essentially have the elementary theory of the
category of sets
(http://www.tac.mta.ca/tac/reprints/articles/11/tr11.pdf) as our set
theory. As noted on page 17 of that paper the category of partially
ordered sets and order preserving maps satisfies all axioms apart from
the axiom of choice. Thus Schroder-Bernstein is independen
t of these axioms apart from choice. The axioms given in the paper are
equivalent to the axioms for a well-pointed topos with a natural number
object. The problem is that in Lawvere's paper, the axiom of choice is
used heavily - for example the proof that 2 is the subobject classifier
relies on choice. I am not sure what conditions should be placed on a
topos to make Schroder-Bernstein hold.
Kind Regards,
Steven Gubkin